Gelfand numbers of embeddings of mixed Besov spaces

Abstract

Gelfand numbers represent a measure for the information complexity which is given by the number of information needed to approximate functions in a subset of a normed space with an error less than ε. More precisely, Gelfand numbers coincide up to the factor 2 with the minimal error e wor (n, Λ all ) which describes the error of the optimal (non-linear) algorithm that is based on n arbitrary linear functionals. This explains the crucial role of Gelfand numbers in the study of approximation problems. Let S t p1,p1 B((0, 1) d ) be the Besov spaces with dominating mixed smoothness on (0, 1)d . In this paper we consider the problem App : 0, 1) d ) and investigate the asymptotic behaviour of Gelfand numbers of this embedding. We shall give the correct order of convergence of Gelfand numbers in almost all cases. In addition we shall compare these results with the known behaviour of approximation numbers which coincide with e wor−lin (n, Λ all ) when we only allow linear algorithms.

Entropy numbers of embeddings of Schatten classes

“…On the application side, Gelfand numbers (of canonical embeddings) naturally appear when considering the problem of optimal recovery of an element f ∈ X from few arbitrary linear samples, where the recovery error is measured in the norm of the codomain space Y , which substantiates their role in the flourishing fields of information-based complexity (see, e.g., [40,41]) and approximation theory [10,12,14,44].…”

Gelfand numbers related to structured sparsity and Besov space embeddings with small mixed smoothness

“…Observe that the decay behaviour in (38) is typical for the univariate situation (d = 1). Usually, in the dvariate setting we encounter asymptotic orders such as n −r (log n) (d−1)η for some η := η(p 0 , q 0 , p 1 , q 1 , r 0 , r 1 ), see Remark 7.9 and [30,7]. The dimension d of the underlying Euclidean space enters the rate of convergence exponentially.…”

Monte Carlo methods for uniform approximation on periodic Sobolev spaces with mixed smoothness

“…Remark 4.10 (Comparison with deterministic approximation). From [11] and Temlyakov [15] we know that for 1 < p ≤ 2 and r > 1 we have…”